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          常用 tkz-euclide 命令（一）——点的定义方法
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    <div class="post-body" itemprop="articleBody"><p>基本格式：</p>
<figure class="highlight latex"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">\begin</span>&#123;tikzpicture&#125;</span><br><span class="line">  <span class="comment">%%% 定义基础点</span></span><br><span class="line">  <span class="keyword">\tkzDefPoint</span>(0,0)&#123;A&#125;</span><br><span class="line">  <span class="keyword">\tkzDefPoint</span>(1,0)&#123;B&#125;</span><br><span class="line">  <span class="comment">%%% 定义各种图形及其它点</span></span><br><span class="line">  <span class="keyword">\tkzDefSquare</span>(A,B) <span class="keyword">\tkzGetPoints</span>&#123;C&#125;&#123;D&#125;</span><br><span class="line">  <span class="comment">%%% 画出所有图形和点</span></span><br><span class="line">  <span class="keyword">\tkzDrawPolygon</span>(A,B,C,D)</span><br><span class="line">  <span class="keyword">\tkzDrawPoints</span>(A,...,D)</span><br><span class="line">  <span class="comment">%%% 标记图形和点</span></span><br><span class="line">  <span class="keyword">\tkzLabelPoints</span>[below](A,B)</span><br><span class="line">  <span class="keyword">\tkzLabelPoints</span>[above](C,D)</span><br><span class="line"><span class="keyword">\end</span>&#123;tikzpicture&#125;</span><br></pre></td></tr></table></figure>
<p>相较于 <code>tikz</code> 宏包的优点：</p>
<ul>
<li>常用的图形都有定义，可以直接画</li>
<li>命令的命名规则比较好记</li>
</ul>
<p>缺点：</p>
<ul>
<li>命名比较繁琐</li>
</ul>
<h2 id="1-定义点的方法"><a class="header-anchor" href="#1-定义点的方法"></a>1. 定义点的方法</h2>
<h3 id="1-1-定义坐标点"><a class="header-anchor" href="#1-1-定义坐标点"></a>1.1. 定义坐标点</h3>
<table>
<thead>
<tr>
<th>方法</th>
<th>命令</th>
</tr>
</thead>
<tbody>
<tr>
<td>直角坐标</td>
<td><code>\tkzDefPoint(&lt;x,y&gt;)&#123;A&#125;</code></td>
</tr>
<tr>
<td>极坐标</td>
<td><code>\tkzDefPoint(&lt;θ:ρ&gt;)&#123;A&#125;</code></td>
</tr>
<tr>
<td>相对坐标</td>
<td><code>\tkzDefShiftPoint[A](&lt;x,y&gt; or &lt;θ:ρ&gt;)&#123;B&#125;</code> 或者 <br /> <code>\tkzDefPoint[shift=&#123;(&lt;x,y&gt; or &lt;θ:ρ&gt;)&#125;]((&lt;x,y&gt; or &lt;θ:ρ&gt;)&#123;B&#125;)</code></td>
</tr>
<tr>
<td>批量定义（直角坐标）</td>
<td><code>\tkzDefPoints&#123;&lt;x1/y1/A,x2/y2/B,...&gt;&#125;</code></td>
</tr>
</tbody>
</table>
<h3 id="1-2-定义圆上的点"><a class="header-anchor" href="#1-2-定义圆上的点"></a>1.2. 定义圆上的点</h3>
<p>命令：<code>\tkzDefPointOnCircle[&lt;option&gt;] \tkzGetPoint&#123;B&#125;</code></p>
<table>
<thead>
<tr>
<th>描述</th>
<th>选项</th>
</tr>
</thead>
<tbody>
<tr>
<td>已知圆上一点</td>
<td><code>through=angle 30 center O point A</code></td>
</tr>
<tr>
<td>已知半径</td>
<td><code>R=angle 30 center O radius \rOA</code></td>
</tr>
</tbody>
</table>
<h3 id="1-3-定义相对点"><a class="header-anchor" href="#1-3-定义相对点"></a>1.3. 定义相对点</h3>
<table>
<thead>
<tr>
<th>方法</th>
<th>命令</th>
</tr>
</thead>
<tbody>
<tr>
<td>中点</td>
<td><code>\tkzDefMidPoint(A,B) \tkzGetPoint&#123;C&#125;</code></td>
</tr>
<tr>
<td>黄金分割点（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>A</mi><mi>C</mi></mrow><mrow><mi>C</mi><mi>B</mi></mrow></mfrac></mstyle><mo>=</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">\dfrac{AC}{CB}=\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">CB</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">φ</span></span></span></span>）</td>
<td><code>\tkzDefGoldenRatio(A,B) \tkzGetPoint&#123;C&#125;</code></td>
</tr>
<tr>
<td>定比分点（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>P</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mi>k</mi><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AP}=k\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
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-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>）</td>
<td><code>\tkzDefPointOnLine[pos=k](A,B)</code></td>
</tr>
<tr>
<td>定比分点（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>P</mi><mo>:</mo><mi>P</mi><mi>B</mi><mo>=</mo><mi>m</mi><mo>:</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">AP:PB=m:n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">PB</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>，即 <br /><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>O</mi><mi>P</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>n</mi><mover accent="true"><mrow><mi>O</mi><mi>A</mi></mrow><mo stretchy="true">→</mo></mover><mo>+</mo><mi>m</mi><mover accent="true"><mrow><mi>O</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\overrightarrow{OP}=\dfrac{n\overrightarrow{OA}+m\overrightarrow{OB}}{n+m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">OP</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
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-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.6517em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8823em;"><span style="top:-2.5193em;"><span class="pstrut" style="height:3.2053em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">m</span></span></span><span style="top:-3.4353em;"><span class="pstrut" style="height:3.2053em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8823em;"><span class="pstrut" style="height:3.2053em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal">A</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">m</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">OB</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>）</td>
<td><code>\tkzDefBarycentricPoint(&lt;A=n,B=m&gt;) \tkzGetPoint&#123;P&#125;</code></td>
</tr>
<tr>
<td>重心坐标</td>
<td><code>\tkzDefBarycentricPoint(&lt;A=α1,B=α2,C=α3,...&gt;) \tkzGetPoint&#123;P&#125;</code></td>
</tr>
<tr>
<td>内/外相似中心</td>
<td><code>\tkzDefSimilitudeCenter[int/ext](O,A)(O',B) \tkzGetPoint&#123;I&#125;</code> 或 <br /> <code>\tkzDefSimilitudeCenter[int/ext,R](O,r)(O',r') \tkzGetPoint&#123;I&#125;</code></td>
</tr>
<tr>
<td>调和分割点（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo separator="true">;</mo><mi>C</mi><mo separator="true">,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(A,B;C,D)=-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>）</td>
<td><code>\tkzDefHarmonic(A,B,k)</code> 或 <br /> <code>\tkzDefHarmonic[ext](A,B,C)</code> 或 <br /> <code>\tkzDefHarmonic[int](A,B,D)</code></td>
</tr>
<tr>
<td>等距点（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mi>A</mi><mo>=</mo><mi>O</mi><mi>B</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">OA=OB=2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">OB</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>）</td>
<td><code>\tkzDefEquiPoints[from=O,dist=2](A,B)</code></td>
</tr>
</tbody>
</table>
<h3 id="1-4-定义几何变换点"><a class="header-anchor" href="#1-4-定义几何变换点"></a>1.4. 定义几何变换点</h3>
<p>命令：<code>\tkzDefPointBy[&lt;option&gt;](P) \tkzGetPoint&#123;Q&#125;</code></p>
<p>批量变换：<code>\tkzDefPointsBy[&lt;option&gt;](M,N,...)&#123;P,Q,...&#125;</code></p>
<p>（如后面的为<code>&#123;&#125;</code>，则缺省值为<code>M',N',...</code>）</p>
<table>
<thead>
<tr>
<th>变换</th>
<th>选项</th>
</tr>
</thead>
<tbody>
<tr>
<td>平移</td>
<td><code>translation=from A to B</code></td>
</tr>
<tr>
<td>位似</td>
<td><code>homothety=center A ratio .5</code></td>
</tr>
<tr>
<td>反射（轴对称）</td>
<td><code>reflection=over A--B</code></td>
</tr>
<tr>
<td>中心对称</td>
<td><code>symmetry=center A</code></td>
</tr>
<tr>
<td>投影</td>
<td><code>projection=onto A--B</code></td>
</tr>
<tr>
<td>旋转（角度）</td>
<td><code>rotation=center O angle 30</code></td>
</tr>
<tr>
<td>旋转（弧度）</td>
<td><code>rotation in rad=center O angle pi/3</code></td>
</tr>
<tr>
<td>旋转（点）</td>
<td><code>rotation with nodes=center O from A to B</code></td>
</tr>
<tr>
<td>反演</td>
<td><code>inversion=center O through A</code></td>
</tr>
<tr>
<td>反演+对称</td>
<td><code>inversion negative=center O through A</code></td>
</tr>
</tbody>
</table>
<h3 id="1-5-定义向量变换点"><a class="header-anchor" href="#1-5-定义向量变换点"></a>1.5. 定义向量变换点</h3>
<p>命令：<code>\tkzDefPointWith[&lt;options&gt;](&lt;A,B&gt;) \tkzGetPoint&#123;C&#125;</code></p>
<table>
<thead>
<tr>
<th>变换</th>
<th>描述</th>
<th>选项</th>
</tr>
</thead>
<tbody>
<tr>
<td>正交</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>C</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mi>K</mi><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AC}=K\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
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-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
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-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
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<td><code>orthogonal</code></td>
</tr>
<tr>
<td>单位正交</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>C</mi><mo>=</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">AC=K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>C</mi></mrow><mo stretchy="true">→</mo></mover><mo>⊥</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AC}\perp\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
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-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>．</td>
<td><code>orthogonal normed</code></td>
</tr>
<tr>
<td>共线</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>C</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mi>K</mi><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AC}=K\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>．</td>
<td><code>linear</code></td>
</tr>
<tr>
<td>单位共线</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>C</mi><mo>=</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">AC=K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>C</mi></mrow><mo stretchy="true">→</mo></mover><mo>∥</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AC}\parallel\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4553em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>．</td>
<td><code>linear normed</code></td>
</tr>
<tr>
<td>共线</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>M</mi><mi>C</mi></mrow><mo stretchy="true">→</mo></mover><mo>=</mo><mi>K</mi><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{MC}=K\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">MC</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>．</td>
<td><code>colinear=at M</code></td>
</tr>
<tr>
<td>单位共线</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>C</mi><mo>=</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">MC=K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">MC</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>M</mi><mi>C</mi></mrow><mo stretchy="true">→</mo></mover><mo>∥</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{MC}\parallel\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4553em;vertical-align:-0.25em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">MC</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>．</td>
<td><code>colinear normed=at M</code></td>
</tr>
<tr>
<td>变换系数</td>
<td>K默认为1（与前面的选项组合使用）</td>
<td><code>K=1</code></td>
</tr>
</tbody>
</table>
<p>获取向量的坐标：<code>\tkzGetVectxy(&lt;A,B&gt;)&#123;&lt;V&gt;&#125;</code>，则向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2053em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.2053em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span class="svg-align" style="top:-3.6833em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span> 的坐标为 (<code>\Vx</code>,<code>\Vy</code>)．</p>
<h3 id="1-6-定义三角形的各中心"><a class="header-anchor" href="#1-6-定义三角形的各中心"></a>1.6. 定义三角形的各中心</h3>
<p>命令：<code>\tkzDefTriangleCenter[&lt;option&gt;](&lt;A,B,C&gt;) \tkzGetPoint&#123;P&#125;</code></p>
<table>
<thead>
<tr>
<th>ETC 编号</th>
<th>名称</th>
<th>描述</th>
<th>选项</th>
</tr>
</thead>
<tbody>
<tr>
<td>X(1)</td>
<td>内心</td>
<td>三条内角平分线的交点</td>
<td><code>in</code></td>
</tr>
<tr>
<td></td>
<td>旁心（与B对应）</td>
<td>两条外角平分线与一条内角平分线的交点</td>
<td><code>ex</code></td>
</tr>
<tr>
<td>X(2)</td>
<td>重心</td>
<td>三条中线的交点</td>
<td><code>centroid</code></td>
</tr>
<tr>
<td>X(3)</td>
<td>外心</td>
<td>三条垂直平分线的交点</td>
<td><code>circum</code></td>
</tr>
<tr>
<td>X(4)</td>
<td>垂心</td>
<td>三条高的交点</td>
<td><code>ortho</code></td>
</tr>
<tr>
<td>X(5)</td>
<td>欧拉圆（九点圆）心</td>
<td>三边的中点、三高的垂足、<br />顶点到垂心的三条线段的中点<br />所在圆的圆心</td>
<td><code>euler</code></td>
</tr>
<tr>
<td>X(6)</td>
<td>类似重心（Lemoine点、莱莫恩点）</td>
<td>重心的等角共轭点</td>
<td><code>symmedian</code> <br /> 或 <code>lemoine</code> <br /> 或 <code>grebe</code></td>
</tr>
<tr>
<td>X(7)</td>
<td>Gergonne点（热尔岗点）</td>
<td>内切点与对应顶点的三条连线的交点</td>
<td><code>gergonne</code></td>
</tr>
<tr>
<td>X(8)</td>
<td>Nagel点（奈格尔点）</td>
<td>旁切点与对应顶点的三条连线的交点</td>
<td><code>nagel</code></td>
</tr>
<tr>
<td>X(9)</td>
<td>mittenpunkt点</td>
<td>旁切点三角形的类似重心</td>
<td><code>mittenpunkt</code></td>
</tr>
<tr>
<td>X(10)</td>
<td>Spieker点（斯俾克心）</td>
<td>中点三角形的内心</td>
<td><code>spieker</code></td>
</tr>
<tr>
<td>X(11)</td>
<td>费尔巴哈点</td>
<td>内切圆与九点圆的切点</td>
<td><code>feuerbach</code></td>
</tr>
</tbody>
</table>
<h3 id="1-7-定义随机点"><a class="header-anchor" href="#1-7-定义随机点"></a>1.7. 定义随机点</h3>
<p>命令：<code>\tkzDefRandPointOn[&lt;local option&gt;]  \tkzGetPoint&#123;P&#125;</code></p>
<table>
<thead>
<tr>
<th>位置</th>
<th>选项</th>
</tr>
</thead>
<tbody>
<tr>
<td>线段</td>
<td><code>segment=A--B</code></td>
</tr>
<tr>
<td>直线</td>
<td><code>line=A--B</code></td>
</tr>
<tr>
<td>矩形</td>
<td><code>rectangle=A and B</code></td>
</tr>
<tr>
<td>圆（已知半径长度）</td>
<td><code>circle=center A radius 2</code></td>
</tr>
<tr>
<td>圆（已知半径线段）</td>
<td><code>circle through=center A through B</code></td>
</tr>
<tr>
<td>圆盘（已知半径线段）</td>
<td><code>disk through=center A through B</code></td>
</tr>
</tbody>
</table>
<h2 id="2-获取点的方法"><a class="header-anchor" href="#2-获取点的方法"></a>2. 获取点的方法</h2>
<p>获取一个点：<code>\tkzGetPoint&#123;A&#125;</code>，默认存储为 <code>tkzPointResult</code></p>
<p>获取多个点：<code>\tkzGetPoints&#123;A&#125;&#123;B&#125;</code>，默认存储为 <code>tkzFirstPointResult</code> 和 <code>tkzSecondPointResult</code></p>
<p>若只获取其中某一个，则可以使用 <code>tkzGetFirstPoint&#123;A&#125;</code> 或 <code>\tkzGetSecondPoint&#123;B&#125;</code></p>
<h2 id="3-绘制点的方法"><a class="header-anchor" href="#3-绘制点的方法"></a>3. 绘制点的方法</h2>
<p>绘制单个点：<code>\tkzDrawPoint[&lt;options&gt;](A)</code></p>
<p>绘制多个点：<code>\tkzDrawPoints[&lt;options&gt;](A,B,C,...)</code></p>
<p>自定义点的样式：<code>\tkzSetUpPoint[&lt;options&gt;]</code></p>
<table>
<thead>
<tr>
<th>样式</th>
<th>选项</th>
</tr>
</thead>
<tbody>
<tr>
<td>形状</td>
<td><code>shape=circle</code>（或 <code>cross</code>、<code>cross out</code>）</td>
</tr>
<tr>
<td>大小</td>
<td><code>size=3</code></td>
</tr>
<tr>
<td>颜色</td>
<td><code>color=black</code></td>
</tr>
<tr>
<td>填充</td>
<td><code>fill=black!50</code></td>
</tr>
</tbody>
</table>
<h2 id="4-标记点的方法"><a class="header-anchor" href="#4-标记点的方法"></a>4. 标记点的方法</h2>
<p>标记单个点：<code>\tkzLabelPoint[&lt;options&gt;](A)&#123;&lt;text support tex&gt;&#125;</code></p>
<p>标记多个点：<code>\tkzLabelPoints[&lt;options&gt;](A,B,C,...)</code></p>
<p>自动选择位置标记多个点：<code>\tkzAutoLabelPoints[center=M, &lt;options&gt;](A,B,C,...)</code></p>
<table>
<thead>
<tr>
<th>描述</th>
<th>选项</th>
</tr>
</thead>
<tbody>
<tr>
<td>位置</td>
<td><code>above/below + left/right</code></td>
</tr>
<tr>
<td>具体位置</td>
<td><code>below right=3pt</code></td>
</tr>
<tr>
<td>字体大小</td>
<td><code>font=\scriptsize</code></td>
</tr>
<tr>
<td>颜色</td>
<td><code>color=black</code></td>
</tr>
</tbody>
</table>

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